SHARYGIN 2019 Problem 3

The rectangle $ABCD$ lies inside a circle. The rays $BA$ and $DA$ meet this circle at points $A_1$ and $A_2$. Let $A_0$ be the midpoint of $A_1A_2$. Points $B_0, C_0, D_0$ are defined similarly. Prove that $A_0C_0 = B_0D_0$.